. 25
( 27)


= Vnm fm , (A.74)
dt m

with a constant matrix V , and initial data fn (0), one takes the Laplace transform to
ζ fn (ζ) ’ Vnm fm (ζ) = fn (0) . (A.75)

This set of algebraic equations can be solved to express fn (ζ) in terms of fn (0).
Inverting the Laplace transform yields the solution in the time domain.
The convolution theorem for Laplace transforms is
t ζ0 +i∞

dt g (t ’ t ) f (t ) = g (ζ) f (ζ) eζt , (A.76)
ζ0 ’i∞

where the integration contour is to the right of any poles of both g (ζ) and f (ζ).
Functional analysis

An important point for applications to physics is that poles in the Laplace trans-
form correspond to exponential time dependence. For example, the function f (t) =
exp (zt) has the transform
f (ζ) = . (A.77)
ζ ’z
More generally, consider a function f (ζ) with N simple poles in ζ:
f (ζ) = , (A.78)
(ζ ’ z1 ) · · · (ζ ’ zN )
where the complex numbers z1 , . . . , zN are all distinct. The inverse transform is
ζ0 +i∞

f (t) = , (A.79)
2πi (ζ ’ z1 ) · · · (ζ ’ zN )
ζ0 ’i∞

where ζ0 > max[Re z1 , . . . , Re zN ]. The contour can be closed by a large semicircle in
the left half plane, and for N > 1 the contribution from the semicircle can be neglected.
The integral is therefore given by the sum of the residues,
ezn t
f (t) = , (A.80)
zn ’ zj
n=1 j=n

which explicitly exhibits f (t) as a sum of exponentials.

A.6 Functional analysis
A.6.1 Linear functionals
In normal usage, a function, e.g. f (x), is a rule assigning a unique value to each value
of its argument. The argument is typically a point in some ¬nite-dimensional space,
e.g. the real numbers R, the complex numbers C, three-dimensional space R3 , etc. The
values of the function are also points in a ¬nite-dimensional space. For example, the
classical electric ¬eld is represented by a function E (r) that assigns a vector”a point
in R3 ”to each position r in R3 .
A rule, X, assigning a value to each point f in an in¬nite-dimensional space M
(which is usually a space of functions) is called a functional and written as X [f ].
The square brackets surrounding the argument are intended to distinguish functionals
from functions of a ¬nite number of variables.
If M is a vector space, e.g. a Hilbert space, then a functional Y [f ] that obeys

Y [±f + βg] = ±Y [f ] + βY [g] , (A.81)

for all scalars ±, β and all functions f, g ∈ M, is called a linear functional. The
family, M , of linear functionals on M is called the dual space of M. The dual space
is also a vector space, with linear combinations of its elements de¬ned by

(±X + βY ) [f ] = ±X [f ] + βY [f ] (A.82)

for all f ∈ M.

A.6.2 Generalized functions
In Section 3.1.2 the de¬nition (3.18) and the rule (3.21) are presented with the cavalier
disregard for mathematical niceties that is customary in physics. There are however
some situations in which more care is required. For these contingencies we brie¬‚y
outline a more respectable treatment. The chief di¬culty is the existence of the in-
tegrals de¬ning the operators s ’∇2 . This problem can be overcome by restricting
the functions • (r) in eqn (3.18) to good functions (Lighthill, 1964, Chap. 2), i.e.
in¬nitely-di¬erentiable functions that fall o¬ faster than any power of |r|. The Fourier
transform of a good function is also a good function, so all of the relevant integrals
exist, as long as s (|k|) does not grow exponentially at large |k|. The examples we need
are all of the form |k| , where ’1 ± 1, so eqns (3.18) and (3.21) are justi¬ed.
For physical applications the really important assumption is that all functions can be
approximated by good functions.
A generalized function is a linear functional, say G [•], de¬ned on the good
functions, i.e.

G [•] is a complex number and G [±• + βψ] = ±G [•] + βG [•] (A.83)

for any scalars ±, β and any good functions •, ψ. A familiar example is the delta
function. The rule
d3 rδ (r ’ R) • (r) = • (R) (A.84)

maps the function • (r) into the single number • (R). In this language, the transverse
delta function ∆⊥ (r ’ r ) is also a generalized function. An alternative terminology,
often found in the mathematical literature, labels good functions as test functions and
generalized functions as distributions.
In quantum ¬eld theory, the notion of a generalized function is extended to linear
functionals sending good functions to operators, i.e. for each good function •,

X [•] is an operator and X [±• + βψ] = ±X [•] + βX [•] . (A.85)

Such functionals are called operator-valued generalized functions. For any density
operator ρ describing a physical state, X [•] de¬nes an ordinary (c-number) generalized
function Xρ [•] by
Xρ [•] = Tr (ρX [•]) . (A.86)

A.7 Improper functions
A.7.1 The Heaviside step function
The step function θ (x) is de¬ned by

1 for x > 0 ,
θ (x) = (A.87)
0 for x < 0 ,

and it has the useful representation
Improper functions

ds e’isx
θ (x) = ’ lim , (A.88)
’0 2πi s + i

which is proved using contour integration.

A.7.2 The Dirac delta function
A Standard properties
(1) If the function f (x) has isolated, simple zeros at the points x1 , x2 , . . . then
δ x ’ xi .
δ (f (x)) = (A.89)
i dx x=xi

The multidimensional generalization of this rule is
δ x ’ xi ,
δ f (x) = (A.90)
‚x x=xi

where x = (x1 , x2 , . . . , xN ), f (x) = (f1 (x) , f2 (x) , . . . , fN (x)),

δ f (x) = δ (f1 (x)) · · · δ (fN (x)) ,
δ x ’ xi = δ x1 ’ xi · · · δ xN ’ xi ,
1 N

the Jacobian ‚f /‚x is the N — N matrix with components ‚fn /‚xm , and xi
satis¬es fn xi = 0, for n = 1, . . . , N .
(2) The derivative of the delta function is de¬ned by

d df
δ (x ’ a) = ’
dxf (x) . (A.92)
dx dx
’∞ x=a

(3) By using contour integration methods one gets
= P ’ iπδ (x) , (A.93)
’0 x + i x
where P is the principal part de¬ned by
∞ ’a ∞
f (x) f (x) f (x)
= lim + . (A.94)
P dx dx dx
x x x
’∞ ’∞ a

(4) The de¬nition of the Fourier transform yields

dtei(ω’ν)t = 2πδ (ω ’ ν) (A.95)

in one dimension, and

d3 rei(k’q)·r = (2π)3 δ (k ’ q) (A.96)

in three dimensions.

(5) The step function satis¬es
θ (x) = δ (x) . (A.97)
(6) The end-point rule is
dxδ (x ’ a) f (x) = f (a) . (A.98)

(7) The three-dimensional delta function δ (r ’ r ) is de¬ned as

δ (r ’ r ) = δ (x ’ x ) δ (y ’ y ) δ (z ’ z ) , (A.99)

and is expressed in polar coordinates by
δ (r ’ r ) = δ (r ’ r ) δ (cos θ ’ cos θ ) δ (φ ’ φ ) . (A.100)
B A special representation of the delta function
In many calculations, particularly in perturbation theory, one encounters functions of
the form
· (ωt)
ξ (ω, t) = , (A.101)
which have the limit
lim ξ (ω, t) = ξ0 δ (ω) , (A.102)

provided that the integral

· (u)
ξ0 = du (A.103)

A.7.3 Integral kernels
The de¬nition of a generalized function as a linear rule assigning a complex number
to each good function can be extended to a linear rule that maps a good function, e.g.
f (t), to another good function g (t). The linear nature of the rule means that it can
always be expressed in the form

g (t) = dt W (t, t ) f (t ) . (A.104)

For a ¬xed value of t, W (t, t ) de¬nes a generalized function of t which is called an
integral kernel. This de¬nition is easily extended to functions of several variables,
e.g. f (r). The delta function, the Heaviside step function, etc. are examples of integral
kernels. An integral kernel is positive de¬nite if

dt f — (t) W (t, t ) f (t )
dt 0 (A.105)

for every good function f (t).
Probability and random variables

A.8 Probability and random variables
A.8.1 Axioms of probability
The abstract de¬nition of probability starts with a set „¦ of events and a probability
function P that assigns a numerical value to every subset of „¦. In principle, „¦ could be
any set, but in practice it is usually a subset of RN or CN , or a subset of the integers.
The essential properties of probabilities are contained in the axioms (Gardiner, 1985,
Chap. 2):

0 for all S ‚ „¦;
(1) P (S)
(2) P („¦) = 1;
(3) if S1 , S2 , . . . is a discrete (countable) collection of nonoverlapping sets, i.e.

Si © Sj = … for i = j , (A.106)

P (S1 ∪ S2 ∪ · · · ) = P (Sj ) . (A.107)

The familiar features 0 P (S) 1, P (…) = 0, and P (S ) = 1 ’ P (S), where S
is the complement of S, are immediate consequences of the axioms. If „¦ is a discrete
(countable) set, then one writes P (x) = P ({x}), where {x} is the set consisting of
the single element x. If „¦ is a continuous (uncountable) set, then it is customary to
introduce a probability density p (x) so that

P (S) = dx p (x) , (A.108)

where dx is the natural volume element on „¦.
If „¦ = Rn , the probability density is a function of n variables: p (x1 , x2 , . . . , xn ).
The marginal distribution of xj is then de¬ned as

dx1 · · · dxj+1 · · ·
pj (xj ) = dxj’1 dxn p (x1 , x2 , . . . , xn ) . (A.109)

The joint probability for two sets S and T is P (S © T ); this is the probability
that an event in S is also in T . This is more often expressed with the notation

P (S, T ) = P (S © T ) , (A.110)

which is used in the text. The conditional probability for S given T is

P (S © T )
P (S, T )
P (S | T ) = = ; (A.111)
P (T ) P (T )

this is the probability that x ∈ S, given that x ∈ T .
¼ Mathematics

The compound probability rule is just eqn (A.111) rewritten as

P (S, T ) = P (S | T ) P (T ) . (A.112)

This can be generalized to joint probabilities for more than two outcomes by applying
it several times, e.g.

P (S, T, R) = P (S | T, R) P (T, R)
= P (S | T, R) P (T | R) P (R) . (A.113)

Dividing both sides by P (R) yields the useful rule

P (S, T | R) = P (S | T, R) P (T | R) . (A.114)

Two sets of events S and T are said to be independent or statistically inde-
pendent if the joint probability is the product of the individual probabilities:

P (S, T ) = P (S) P (T ) . (A.115)

A.8.2 Random variables
A random variable X is a function X (x) de¬ned on the event space „¦. The function
can take on values in „¦ or in some other set. For example, if „¦ = R, then X (t) could
be a complex number or an integer. The average value of a random variable is

X= dx p (x) X (x) . (A.116)

If the function X does take on values in „¦, and is one“one, i.e. X (x1 ) = X (x2 ) implies
x1 = x2 , then the distinction between X (x) and x is often ignored.
Appendix B
Classical electrodynamics

B.1 Maxwell™s equations
In SI units the microscopic form of Maxwell™s equations is
∇·E = , (B.1)
∇ — B = µ0 j + , (B.2)
∇—E =’ , (B.3)
∇· B = 0. (B.4)

The homogeneous equations (B.3) and (B.4) are identically satis¬ed by introducing
the scalar potential • and the vector potential A (r) and setting
B = ∇ — A,
E=’ ’ ∇• .
A further consequence of this representation is that eqn (B.1) becomes the Poisson
∇2 • = ’ , (B.6)
which has the Coulomb potential as its solution.
The vector and scalar potentials A and • are not unique. The same electric and
magnetic ¬elds are produced by the new potentials A and • de¬ned by a gauge

A ’ A = A + ∇χ , (B.7)
• ’ • = • ’ ‚χ/‚t , (B.8)

where χ (r, t) is any di¬erentiable, real function. This is called gauge invariance.
This property can be exploited to choose the gauge that is most convenient for the
problem at hand. For example, it is always possible to perform a gauge transformation
such that the new potentials satisfy ∇ · A = 0 and • = ¦, where ¦ is a solution of
eqn (B.6). This is called the Coulomb gauge, since • = ¦ is the Coulomb potential,
or the radiation gauge, since the vector potential is transverse (Jackson, 1999, Sec.
¾ Classical electrodynamics

The ¬‚ow of energy in the ¬eld is described by the continuity equation (Poynting™s
‚u (r, t)
+ ∇ · S (r, t) = 0 , (B.9)
u (r, t) = E 2 (r, t) + B (r, t) (B.10)
2 2µ0
is the electromagnetic energy density, and the Poynting vector
E —B (B.11)
is the energy ¬‚ux.

B.2 Electrodynamics in the frequency domain
It is often useful to describe the ¬eld in terms of its frequency and/or wavevector con-
tent. Let F (r,t) be a real function representing any of the components of E, B, or A .
Under the conventions established in Appendix A.4, the four-dimensional (frequency
and wavevector) Fourier transform of F (r, t) is

dte’i(k·r’ωt) F (r, t) ,
d3 r
F (k, ω) = (B.12)

and the inverse transform is

d3 k dω
F (k, ω) ei(k·r’ωt) .
F (r, t) = (B.13)
3 2π
(2π) ’∞

According to eqn (A.52) the reality of F (r, t) imposes the conditions

F — (k, ω) = F (’k, ’ω) . (B.14)

For many applications it is also useful to consider the temporal Fourier transform at
a ¬xed position r:

dteiωt F (r, t) ,
F (r, ω) = (B.15)

with the inverse transform

F (r, ω) e’iωt .
F (r, t) = (B.16)


The function F (r, ω) satis¬es

F — (r, ω) = F (r, ’ω) . (B.17)
The quantity F (+) (r, ω) is called the power spectrum of F ; it can be used to
de¬ne an average frequency, ω0 , by
Wave equations

∞ dω 2
d3 r F (+) (r, ω) ω
’∞ 2π
ω0 = ω = . (B.18)
∞ dω 2
d3 r F (+) (r, ω)
’∞ 2π

The frequency spread of the ¬eld is characterized by the rms deviation ∆ω”the
frequency or spectral width”de¬ned by
∞ dω 2 2
(r, ω) (ω ’
d3 r (+)
’∞ 2π F ω0 )
2 2
(∆ω) = (ω ’ ω0 ) = . (B.19)
∞ 2
d3 r ’∞ dω F (+) (r, ω)

The average wavevector k0 and deviation ∆k are similarly de¬ned by
∞ dω 2
d3 k
F (+) (k, ω) k
’∞ 2π
k0 = k = , (B.20)
∞ dω 2
d3 k
F (+) (k, ω)
’∞ 2π

∞ dω 2
d3 k 2
(k, ω) (k ’
’∞ 2π F k0 )
2 2
(∆k) = (k ’ k0 ) = . (B.21)
∞ dω 2
d3 k
F (+) (k, ω)
(2π)3 ’∞ 2π

B.3 Wave equations
The microscopic Maxwell equations (B.1)“(B.4) can be replaced by two second-order
wave equations for E and B:

1 ‚2 ‚j 1
∇2 ’ E = µ0 + ∇ρ , (B.22)
2 ‚t2
c ‚t 0
∇2 ’ 2 2 B = ’µ0 ∇ — j , (B.23)
c ‚t

and the ¬rst-order equations for the transverse vector potential A can be combined
to yield the wave equation

1 ‚2
A = ’µ0 j⊥ ,
∇2 ’ (B.24)
c2 ‚t2

where j⊥ is the transverse part (see Section 2.1.1-B) of the current.

B.3.1 Propagation in the vacuum
Since the B ¬eld and the transverse part of the E ¬eld are derived from the vector
potential, we can concentrate on the wave equation (B.24) for the vector potential. In
the vacuum case (j = 0), A satis¬es

1 ‚2
∇2 ’ A (r, t) = 0 , (B.25)
c2 ‚t2

and the transversality condition ∇ · A = 0.
Classical electrodynamics

The general solution of the wave equation can be obtained by a four-dimensional
Fourier transform, which yields

’k + 2 A (k, ω) = 0 .

The solution of the last equation is

A (k, ω) = A(+) (k) 2πδ (ω ’ ck) + A(’) (k) 2πδ (ω + ck) , (B.27)

and the reality of A (r, t) requires

A(+) (k) = A(’) (’k) . (B.28)

The inverse transform yields the general solution in (r, t)-space as

A (r, t) = A(+) (r, t) + A(’) (r, t) , (B.29)


d3 k
A 3A (k) ei(k·r’ωk t) = A(’) (r, t)
(+) (+)
(r, t) = , (B.30)
and ωk = ck.
The relation between the E and B ¬elds and the vector potential can be used to
express them in the same way. In k-space

E (+) (k) = iωk A(+) (k) , B(+) (k) = ik — A(+) (k) , (B.31)

and in (r, t)-space
E (r, t) = E (+) (r, t) + E (’) (r, t) ,
B (r, t) = B(+) (r, t) + B(’) (r, t) ,

d3 k
3 iωk A (k) ei(k·r’ωk t) = E (’) (r, t)
E (+) (+)
(r, t) = ,

d3 k
B(+) (r, t) = 3 ik — A (k) ei(k·r’ωk t) = B(’) (r, t)
B.3.2 Linear and circular polarization
The forms in eqns (B.32) and (B.33) are valid for any real vector solutions of the wave
equation, but we are only interested in transverse ¬elds, e.g. A (r, t) should satisfy
∇ · A (r, t) = 0, as well as the wave equation. In k-space the transversality condition,
k · A(+) (k) = 0, requires A(+) (k) to lie in the plane perpendicular to the direction of
the k-vector. We choose two unit vectors e1 (k) and e2 (k) such that e1 (k) , e2 (k) , k
form a right-handed coordinate system, where k = k/k is the unit vector along the
propagation direction. The unit vectors e1 and e2 are called polarization vectors.
Wave equations

Since an arbitrary vector can be expanded in the basis e1 (k) , e2 (k) , k , the three
unit vectors satisfy the completeness relation

esi esj + ki kj = δij , (B.34)

as well as the conditions

k · es (k) = 0 , (B.35)
es (k) · es (k) = δss , (B.36)
e1 — e2 = k (et cycl) , (B.37)

where s, s = 1, 2. The vector A(+) (k) can therefore be expanded as

A(+) (k) = A(+) (k) es (k) , (B.38)

where the polarization components As (k) are de¬ned by

A(+) (k) = es (k) · A(+) (k) . (B.39)

The general transverse solution of the wave equation is therefore given by eqn (B.29)
d3 k
A (r, t) = A(+) (k) es (k) ei(k·r’ωk t) .
(2π) s
Each plane-wave contribution to the solution of the wave equation, say for E and
B, has the form
E = Re (E1 e1 + E2 e2 ) ei(k·r’ωk t) , (B.41)
B= k—E, (B.42)
where E1 and E2 are complex scalar amplitudes, and e1 and e2 are real polarization
vectors k. If the phases of E1 and E2 are equal, the ¬eld is linearly polarized. If the
phases are di¬erent, the ¬eld is elliptically polarized. In general, the time-averaged
Poynting vector is

1 1
Re E — B— =
0 2 2
|E1 | + |E2 |
S= k, (B.43)
µ0 2 µ0

so the intensity is given by
0 2 2
I = |S| = c |E1 | + |E2 | . (B.44)
If the two phases di¬er by 90—¦ , then

E1 e1 + E2 e2 = E0 (e1 ± ie2 ) , (B.45)
Classical electrodynamics

where E0 is real. The ¬eld is then said to be circularly polarized. In this case it is
useful to introduce the complex unit vectors
es = √ (e1 + ise2 ) , (B.46)
where s = ±1. The complex vectors satisfy the (hermitian) orthogonality relation
e— · es = δss , (B.47)

and the completeness relation
esi e— + ki kj = δij . (B.48)

The transversality, orthogonality, and completeness properties of the linear and circular
polarization vectors are both incorporated in the relations
k · es (k) = 0 (transversality) ,
e— (k) · es (k) = δss (orthonormality) ,
esi (k) e— (k) = δij ’ ki kj (completeness) .

Note that the completeness relation can also be written as
esi (k) e— (k) = ∆⊥ (k) , (B.50)
sj ij

where ∆⊥ (k) is the Fourier transform of the transverse delta function. The general
solution (B.40) has the same form as for linear polarizations, but the polarization
component is now given by
A(+) (k) = e— (k) · A(+) (k) . (B.51)
s s

In addition to eqn (B.49), the circular polarization vectors satisfy
k — es = ’ises , (B.52)
es — e— = ’isδss k , (B.53)

where s, s = ±1.
The linear polarization basis for a given k-vector is not uniquely de¬ned, since
a new basis de¬ned by a rotation around the k-direction also forms a right-handed
coordinate system. It is therefore useful to consider the transformation properties of the
polarization basis. Let ‘ be the rotation angle around k; then the linear polarization
vectors transform by
e1 = e1 cos ‘ + e2 sin ‘ ,
e2 = ’e1 sin ‘ + e2 cos ‘ ,
which implies
es = e1 + ise2 = e’is‘ es . (B.55)
When viewed at a ¬xed point in space by an observer facing into the propagation
direction of the wave (toward the source), the unit vector e+ (e’ ) describes a phasor
Wave equations

rotating counterclockwise (clockwise). In the traditional terminology of optics and
spectroscopy, e+ (e’ ) is said to be left (right) circularly polarized. In the ¬elds of
quantum electronics and laser physics, the observer is assumed to be facing along the
propagation direction (away from the source), so the sense of rotation is reversed.
In this convention e+ (e’ ) is said to be right (left) circularly polarized. In more
modern language e+ (e’ ) is said to have positive (negative) helicity (Jackson, 1999,
Sec. 7.2).
For a plane wave with propagation vector k, there are two amplitudes Es (k), where
for circular (linear) polarization s = ±1 (s = 1, 2). The general vacuum solution can be
expressed as a superposition of plane waves. In this context it is customary to change
the notation by setting
Es (k) = 2i ±s (k) , (B.56)
where the is introduced only to guarantee that |±s (k)| is a density in k-space,
i.e. the new amplitude ±s (k) has dimensions L3/2 . This yields the Fourier integral

d3 k ωk
E (+) (r, t) = i ±s (k) es (k) ei(k·r’ωk t) . (B.57)
3 20
(2π) s

The Fourier integral representation is often replaced by a discrete Fourier series:

E (+) (r, t) = ±ks eks ei(k·r’ωk t) , (B.58)
2 0V

where eks = es (k), ±ks = ±s (k) / V , and V is the volume of the imaginary cube
used to de¬ne the discrete Fourier series.

B.3.3 Spatial inversion and time reversal
Maxwell™s equations are invariant under the discrete transformations

r ’ ’ r (spatial inversion or parity transformation) (B.59)

t ’ ’t (time reversal) , (B.60)
as well as all continuous Lorentz transformations (Jackson, 1999, Sec. 6.10). The phys-
ical meaning of spatial inversion is as follows. If a system of charges and ¬elds evolves
from an initial to a ¬nal state, then the spatially-inverted initial state will evolve
into the spatially-inverted ¬nal state. Time-reversal invariance means that the time-
reversed ¬nal state will evolve into the time-reversed initial state.
For any physical quantity X, let X ’ X P and X ’ X T denote the transforma-
tions for spatial inversion and time reversal respectively. The invariance of Maxwell™s
equations under spatial inversion is achieved by the transformation rules

ρP (r, t) = ρ (’r, t) , jP (r, t) = ’j (’r, t) , (B.61)
Classical electrodynamics

E P (r, t) = ’E (’r, t) , (B.62)
BP (r, t) = B (’r, t) . (B.63)

Thus the current density and the electric ¬eld have odd parity, and the charge density
and the magnetic ¬eld have even parity. Vectors with odd parity are called polar
vectors, and those with even parity are called axial vectors, so E is a polar vector and
B is an axial vector.
Time-reversal invariance is guaranteed by

ρT (r, t) = ρ (r, ’t) , jT (r, t) = ’j (r, ’t) , (B.64)

E T (r, t) = E (r, ’t) , (B.65)
BT (r, t) = ’B (r, ’t) . (B.66)

As a consequence of these rules, the energy density and Poynting vector satisfy

uP (r, t) = u (’r, t) , SP (r, t) = ’S (’r, t) ,
uT (r, t) = u (r, ’t) , ST (r, t) = ’S (r, ’t) .

For many applications, e.g. to scattering problems, it is more useful to work out
the transformation laws for the amplitudes in a plane-wave expansion of the ¬eld. We
begin by using eqn (B.58) to express the two sides of eqn (B.62) as

ωk P
E P (r, t) = ±ks eks ei(k·r’ωk t) + CC
i (B.68)
2 0V

’E (’r, t) = ’ ±ks eks ei(’k·r’ωk t) + CC .
i (B.69)
2 0V

Changing k to ’k in the last result and equating the coe¬cients of corresponding
plane waves yields
±P eks = ’ ±’k,s e’k,s . (B.70)
s s

In order to proceed, we need to relate the polarization vectors for k and ’k. As
a shorthand notation, set es = eks , es = e’k,s , and k = ’k. The vectors es lie in
the same plane as the vectors es , so they can be expressed as linear combinations of
e1 and e2 . After imposing the conditions (B.35)“(B.37) on the basis {e1 , e2 , k }, the
relation between the two basis sets must have the form
e1 = e1 cos ‘ + e2 sin ‘ ,
e2 = e1 sin ‘ ’ e2 cos ‘ .

The transformation matrices in eqns (B.54) and (B.71) represent proper and improper
rotations respectively. The improper rotation in eqn (B.71) can be expressed as the
product of a proper rotation and a re¬‚ection through some line in the plane orthogonal
Planar cavity

to k. Since the polarization basis can be freely chosen, it is convenient to establish a
convention by setting ‘ = 0, i.e.

e’k,1 = ek1 , e’k,2 = ’ek2 . (B.72)

For the circular polarization basis, with s = ±, this rule takes the equivalent forms

e’k,s = e— ,
e’k,’s = eks .

The transformation law derived by applying this rule to eqn (B.70) is

±P = ’±’k,’s (s = ±) , (B.74)

which relates the amplitude for a given wavevector and circular polarization to the
amplitude for the opposite wavevector and opposite circular polarization. For the linear
polarization basis the corresponding result is

±P = ’±’k,1 ,
±P = ±’k,2 .

Turning next to time reversal, we express the right side of eqn (B.65) as

ωk ωk — — ’i(k·r+ωk t)
E (r, ’t) = ±ks eks ei(k·r+ωk t) ’
i i ±ee , (B.76)
2 0 V ks ks
2 0V
ks ks

and again change the summation variable by k ’ ’k. This is to be compared to the
expansion for E T (r, t), which is given by eqn (B.68) with ±P replaced by ±T . The
ks ks
result is
±— e—
±T eks = ’ ’k,s ’k,s . (B.77)
s s

The circular polarization vectors satisfy e— —
’k,s = ek,’s = ek,s , so the transformation
law in this basis is
±T = ’±— ’k,s . (B.78)

Thus for time reversal the amplitude for (k, s) is related to the conjugate of the
amplitude for (’k, s). The wavevector is reversed, but the circular polarization is
unchanged. For the linear basis the result is

±T = ’±—
’k,1 , (B.79)

±T = ±—
’k,2 . (B.80)

B.4 Planar cavity
A limiting case of the rectangular cavity discussed in Section 2.1.1 is the planar cavity,
L. In most applications, only the limit L ’ ∞ will
with L1 = L2 = L and L3 = d
be relevant, so the only physically meaningful boundary conditions are those at the
¼ Classical electrodynamics

planes z = 0 and z = d. Periodic boundary conditions can be used at the other faces
of the cavity. Thus the ansatz for the solution is E = eiq·r U (z), where q = (kx , ky )
is the transverse part of the wavevector k. Inserting this into eqns (2.11), (2.1), and
(2.13) leads to the mode functions E qns . For n = 0 there is only one polarization:

E q0 = √ eiq·r uz . (B.81)
For n 1 there are two polarizations, the P polarization in the (q, uz )-plane and the
orthogonal S polarization along uz — q:

2 n»qn q
E qn1 = eiq·r sin (kz z) q + i cos (kz z) uz , (B.82)
L2 d 2d kz

2 iq·r
E qn2 = sin (kz z) uz — q ,
e (B.83)
L2 d
where »qn = 2πc/ωqn . The mode frequency is

q 2 + (nπ/d)2 ,
ωqn = c (B.84)

and the expansion of a general real ¬eld is
∞ Cn
E (r) = i aqns E qns (z) eiq·r ’ CC , (B.85)
q n=0 s=1

where C0 = 1 and Cn = 2 for n 1.

B.5 Macroscopic Maxwell equations
The macroscopic Maxwell equations are given by (Jackson, 1999, Sec. 6.1)

∇ · D (r, t) = ρ (r, t) , (B.86)
‚D (r, t)
∇ — H (r, t) = J (r, t) + , (B.87)
‚B (r, t)
∇ — E (r, t) = ’ , (B.88)
∇ · B (r, t) = 0 , (B.89)
D (r, t) = 0 E (r, t) + P (r, t) , (B.90)
H (r, t) = B (r, t) ’ M (r, t) . (B.91)
In these equations ρ and J respectively represent the charge density and current
density of the free charges, P is the polarization density (density of the electric
dipole moment), M is the magnetization (density of the magnetic dipole moment),
D is the displacement ¬eld, and H is the magnetic ¬eld.
Macroscopic Maxwell equations

After Fourier transforming in r and t, Maxwell™s equations reduce to the algebraic

k · D (k, ω) = ’iρ (k, ω) , (B.92)
k — H (k, ω) = ’iJ (k, ω) ’ ωD (k, ω) , (B.93)
k — E (k, ω) = ωB (k, ω) , (B.94)
k · B (k, ω) = 0 , (B.95)
0E (k, ω) + P (k, ω) ,
D (k, ω) = (B.96)
H (k, ω) = B (k, ω) ’ M (k, ω) . (B.97)

The microscopic Poynting™s theorem (B.9) is replaced by

‚D ‚B
E· +H· +∇· S = 0, (B.98)
‚t ‚t

where S = E — H (Jackson, 1999, Sec. 6.7).
For a nondispersive medium, i.e.

Di (r, t) = ij Ej (r, t) , Bi (r, t) = µij Hj (r, t) , (B.99)

where and µij are constant tensors, eqn (B.98) takes the form

‚u (r, t)
+ ∇ · S (r, t) = 0 , (B.100)
with the energy density

{E · D + B · H}
u= (B.101)
Ei ij Ej + Bi µ’1 Bj .
= (B.102)
The most important materials for quantum optics are nonmagnetic dielectrics with
µij (ω) = µ0 δij . In this case eqns (B.86)“(B.91) can be converted into a wave equation
for the transverse part of the electric ¬eld:

1 ‚2 ‚2 ‚

= µ0 2 P ⊥ + µ0 J⊥ .
∇’2 2 E
c ‚t ‚t ‚t

B.5.1 Dispersive linear media
We consider a medium which interacts weakly with external ¬elds. This can happen
either because the ¬elds themselves are weak or because the e¬ective coupling constants
are small. In general, the polarization and magnetization at a space“time point x =
(r, t) can depend on the action of the ¬eld at earlier times and at distant points
¾ Classical electrodynamics

in space. Combining this with the weak interaction assumption leads to the linear
constitutive equations (Jackson, 1999, p. 14)

Pi (r, t) = dt χij (r ’ r , t ’ t ) Ej (r , t ) ,
d3 r (B.104)

Mi (r, t) = dt ξij (r ’ r , t ’ t ) Hj (r , t ) ,
d3 r (B.105)

(1) (1)
where χij (r ’ r , t ’ t ) and ξij (r ’ r , t ’ t ) are respectively the (linear) electric
and magnetic susceptibility tensors. Thus the relation between the polarization P (r, t)
(magnetization M (r, t)) and the ¬eld E (r, t) (H (r, t)) is nonlocal in both space and
time. The principle of causality prohibits P (r, t) (M (r, t)) from depending on the
¬eld E (r, t ) (H (r, t )) at later times, t > t, so the susceptibilities must satisfy
χij (r ’ r , t ’ t ) = 0 ,
for t > t . (B.106)
(r ’ r , t ’ t ) = 0

This leads to the famous Kramers“Kronig relations (Jackson, 1999, Sec. 7.10).
The four-dimensional convolution theorem, obtained by combining eqns (A.55) and
(A.57), allows eqns (B.104) and (B.105) to be recast in Fourier space as
Pi (k, ω) = (k, ω) Ej (k, ω) ,
0 χij (B.107)
Mi (k, ω) = ξij (k, ω) Ej (k, ω) . (B.108)

Combining these relations with the de¬nitions (B.90) and (B.91) produces

Di (k, ω) = (k, ω) Ej (k, ω) (B.109)

Bi (k, ω) = µij (k, ω) Hj (k, ω) , (B.110)
(k, ω) ≡ δij + χij (k, ω) (B.111)
ij 0

µij (k, ω) ≡ µ0 δij + ξij (k, ω) (B.112)

are respectively the (electric) permittivity tensor and the (magnetic) permeabil-
ity tensor. The classical ¬elds, the polarization, the magnetization, and the (space“
time) susceptibilities are all real; therefore, the Fourier transforms satisfy

P — (k, ω) = P (’k, ’ω) , E — (k, ω) = E (’k, ’ω) ,
M— (k, ω) = M (’k, ’ω) , B— (k, ω) = B (’k, ’ω) , (B.113)
(1)— (1) (1)— (1)
(k, ω) = χij (’k, ’ω) , ξij (k, ω) = ξij (’k, ’ω) .
(1) (1)
The dependence of χij (k, ω) and ξij (k, ω) on k is called spatial dispersion, and
the dependence on ω is called frequency dispersion. Interactions between atoms at
Macroscopic Maxwell equations

di¬erent points in the medium can cause the polarization at a point r to depend on the
¬eld in a neighborhood of r, de¬ned by a spatial correlation length as . In gases, liquids,
and disordered solids as is of the order of the interatomic spacing, which is generally
very small compared to vacuum optical wavelengths »0 . Thus the polarization at r
can be treated as depending only on the ¬eld at r. Since the medium is assumed to
(1) (1)
be spatially homogeneous, this means that χij (r ’ r , t ’ t ) = χij (t ’ t ) δ (r ’ r ),
(1) (1)
which is equivalent to χij (k, ω) = χij (ω). Similar relations hold for the magnetic
susceptibility. These three types of media are also isotropic (rotationally symmetric),
so the tensor quantities can be replaced by scalars which depend only on ω:

(k, ω) ’ (ω) δij ,
1 + χ(1) (ω) ,
(ω) = 0

µij (k, ω) ’ µ (ω) δij ,
µ (ω) = µ0 1 + ξ (1) (ω) .

Using eqn (B.114) in eqn (B.109) and transforming back to position space produces
the useful relation
D (r, ω) = (ω) E (r, ω) . (B.116)
For crystalline solids, rotational symmetry is replaced by symmetry under the
crystal group, and the tensor character of the susceptibilities cannot be ignored. In
this case as is the lattice spacing, so the ratio as /»0 is still small, but spatial dispersion
cannot always be neglected. The reason is that the relevant parameter is n (ω0 ) as /»0 ,
where n (ω0 ) is the index of refraction at the frequency ω0 = 2πc/»0 . Thus spatial
dispersion can be signi¬cant if the index is large.
In a rare stroke of good fortune, the crystals of interest for quantum optics satisfy
the condition for weak spatial dispersion, n (ω0 ) as /»0 1 (Agranovich and Ginzburg,
1984); therefore, we can still use a permittivity tensor that only depends on frequency:

(k, ω) ’ (ω) . (B.117)
ij ij

For most applications of quantum optics, we can also assume that the permittivity
tensor is symmetrical: ij (ω) = ji (ω). Physically this means that the crystal is both
transparent and non-gyrotropic (not optically active) (Agranovich and Ginzburg, 1984,
Chap. 1). We also assume the existence of the inverse tensor ’1 ij .
There are other situations, e.g. propagation in a plasma exposed to an external
magnetic ¬eld, that require the full tensors ij (k, ω) and µij (k, ω) depending on k
(Pines, 1963, Chaps 3 and 4; Ginzburg, 1970, Sec. 1.2). For nearly all applications of
quantum optics, we can neglect spatial dispersion and assume the forms (B.114) or
(B.117) for the permittivity tensor.
The inertia of the charges and currents in the medium, together with dissipative
e¬ects, imply that the medium cannot respond instantaneously to changes in the ¬eld
at a given point r. Thus the polarization at the position r and time t will in general
depend on the ¬eld at earlier times t < t. Since the response times of gases, liquids,
Classical electrodynamics

and solids exhibit considerable variation, it is not generally possible to ignore frequency

B.5.2 Isotropic linear dielectrics
Here we assume that µij (ω) = µ0 δij , so that H = B/µ0 , and set (ω) = δij (ω)
and ρ = J = 0 in eqns (B.92)“(B.95) to get

k · E (k, ω) = 0 , (B.118)
k — B (k, ω) = ’ r (ω) E (k, ω) , (B.119)
k — E (k, ω) = ωB (k, ω) , (B.120)
k · B (k, ω) = 0 , (B.121)

where r (ω) = (ω) / 0 is the relative permittivity. The ¬nal equation follows from
eqn (B.120), and eliminating B between eqn (B.119) and eqn (B.120) leads to

k — [k — E (k, ω)] = ’ 2 (ω) E (k, ω) . (B.122)
The identity a — (b — c) = (a · c) b ’ (a · b) c, together with eqn (B.118), reduces this
ω2 2
n (ω) ’ k 2 E (k, ω) = 0 , (B.123)
where n (ω) = r (ω) is the index of refraction. In general r (ω) can be complex,
corresponding to absorption or gain at particular frequencies (Jackson, 1999, Chap. 7),
but for frequencies in the transparent part of the spectrum r (ω) is real and positive.
The relation E = ’‚A/‚t implies E (k, ω) = iωA (k, ω), so the vector potential
satis¬es the same equation

ω2 2
n (ω) ’ k 2 A (k, ω) = 0 . (B.124)

For a transparent medium the general transverse solution of eqn (B.124) is

A— (’k) e— (’k) δ (ω + ω (k)) ,
A (k, ω) = As (k) es (k) δ (ω ’ ω (k)) + s s
s s
where ω (k) is a positive, real solution of the dispersion relation

ωn (ω) = ck . (B.126)

Thus the fundamental plane-wave solution in position“time is

ei(k·r’ω(k)t) es (k) , (B.127)

and the positive-frequency part has the general form
Macroscopic Maxwell equations

d3 k
A As (k) es (k) ei(k·r’ω(k)t)
(r, t) = (B.128)
(2π) s

for the vector potential, and
d3 k
E As (k) es (k) ei(k·r’ω(k)t)
(r, t) = i 3 ω (k) (B.129)
(2π) s

for the electric ¬eld.
B.5.3 Anisotropic linear dielectrics
We again assume that µij (ω) = µ0 δij and set ρ = J = 0 in eqns (B.92)“(B.95), but
we drop the assumption ij (ω) = δij (ω). In this case E and D are not necessarily
parallel, so we combine eqn (B.93) with eqn (B.94) to get
k 2 ∆⊥ Ej = µ0 ω 2 Di . (B.130)

In the following we will use a matrix notation in which a second-rank tensor Xij is

represented as a 3 — 3 matrix X and a vector V = V1 ux + V2 uy + V3 uz is represented
by column or row matrices according to the convention

’ ⎝ 1⎠ ’ T ’
V = V2 , V = (V1 , V2 , V3 ) . (B.131)
The polarization properties of the solution are best described in terms of D, since

eqn (B.92) guarantees that it is orthogonal to k. Thus we solve eqn (B.109) for E and
substitute the result into the left side of eqn (B.130) to ¬nd
’’ ← 1 ’ ’1 ’
’ ’ ← 1 ’ ’1 ← ’
’ ’’
k 2 ∆ ⊥ E = k 2 ∆ ⊥ [← r ] D = k 2 ∆ ⊥ [← r ] ∆ ⊥ D , (B.132)
0 0

where ( r )ij (ω) = (ω) / 0 is the relative permittivity tensor. The last form

depends on the fact that D is transverse. Putting this together with the right side of
eqn (B.130) yields
’’ ω2 ’’
S D = 2 2D, (B.133)
where the transverse impermeability tensor,1

’ ←’ ’1 ←

S k, ω = ∆ ⊥ k [← r (ω)] ∆ ⊥ k ,
’ (B.134)
depends on the frequency ω and the unit vector k = k/k along the propagation vector.
←’ ’

The real, symmetric matrix S annihilates k :
S k = 0, (B.135)

so S has one eigenvalue zero, corresponding to the eigenvector k. From eqn (B.133),

it is clear that the transverse vector D is one of the remaining two eigenvectors that
are orthogonal to k.
1 This is a slight modi¬cation of the approach found in Yariv and Yeh (1984, Chap. 4).
Classical electrodynamics

If ω lies in the transparent region for the crystal, the tensor ← r (ω) is positive

de¬nite, so that the nonzero eigenvalues of S are positive. We write the positive
eigenvalues as 1/n2 , so that the corresponding eigenvectors satisfy

’’ 1’
S µ s = 2 ’ s (s = 1, 2) .
µ (B.136)

’ ’

If D is parallel to an eigenvector, i.e. D = Ys ’ s , one ¬nds the dispersion relation


c2 k 2 = ω 2 n2 (ω) ; (B.137)

in other words, ns is the index of refraction associated with the eigenpolarization
’ (k). Since the matrix ← depends on the direction of propagation k, the indices

’ S
ns (ω) generally also depend on k. In order to simplify the notation, this dependence is
not indicated explicitly, e.g. as ns ω, k , but is implicitly indicated by the dependence
of the refractive index on the polarization index. An incident wave with propagation
vector k exhibits birefringence, i.e. it produces two refracted waves corresponding
to the two phase velocities c/n1 and c/n2 . Since ’ s (k) is real, the eigenpolarizations

are linear, and they can be normalized so that
’ T (k) ’ (k) = µ (k) · µ (k) = δ .
’ ’
µs µs (B.138)
s s ss

Radiation is described by the transverse part of the electric ¬eld, and for the special

solution D = Ys ’ s the transverse electric ¬eld in (k, ω)-space is

’ 1← ’
’’ Ys ’ ’,
E s (k, ω) = S Ys µ s = µs (B.139)
0 ns

where ωs (k) is a solution of eqn (B.137) and ns (k) ≡ ns (ωs (k)). The general space“
time solution,
’ (+)
’ d3 k
1 Ys (k) ’
’ (k) ei(k·r’ωs (k)t) ,
E (r, t) = µs (B.140)
3 2 (k)
0 s=1

is a superposition of elliptically-polarized waves with axes that rotate as the wave
propagates through the crystal. If only one polarization is present, e.g. Y2 (k) = 0, each
wave is linearly polarized, and the polarization direction is preserved in propagation.
It is customary and useful to get a representation similar to eqn (B.57) for the
isotropic problem by setting

0 ωs (k)
Ys (k) = ins (k) ±s (k) , (B.141)
so that the transverse part of the electric ¬eld is
d3 k ωs (k)
E ± (k) µs (k) ei(k·r’ωs (k)t) .
(r, t) = i (B.142)
2 (k) s
3 2 0 ns
(2π) s=1
Macroscopic Maxwell equations

The corresponding expansion using box-normalized plane waves is

E ⊥ (r, t) = i 2 ±ks µks e
i(k·r’ωks t)
, (B.143)
2 0 V nks

where ωks = ωs (k), nks = ns (k), µks = µs (k), and ±ks = ±s (k) / V .
In the presence of sources the coe¬cients are time dependent:
d3 k ωs (k)
E ±s (k, t) µs (k) ei(k·r’ωs (k)t) ,
(r, t) = i (B.144)
3 2 0 n2 (k)
(2π) s

E (+) (r, t) = i ±ks (t) µks ei(k·r’ωks t) . (B.145)
2 0 V n2

For ¬elds satisfying eqns (3.107) and (3.120), the argument used for an isotropic
medium can be applied to the present case to derive the expressions

d3 k 2
U= ωs (k) |±s (k, t)| (B.146)
(2π) s

U= ωks |±s (k, t)| (B.147)

for the energy in the electromagnetic ¬eld.

A Uniaxial crystals
The analysis sketched above is valid for general crystals, but there is one case of
special interest for applications. A crystal is uniaxial if it exhibits threefold, fourfold,
or sixfold symmetry under rotations in the plane perpendicular to a distinguished
axis, which we take as the z-axis. The x- and y-axes can be any two orthogonal
lines in the perpendicular plane. In general, the permittivity tensor is diagonal”
with diagonal elements x , y , z ”in the crystal-axis coordinates, but the symmetry
under rotations around the z-axis implies that x = y . We set x = y = ⊥ , but in
general ⊥ = z . In these coordinates, the unit vector along the propagation direction
is k = k/k = (sin θ cos φ, sin θ sin φ, cos θ), where θ and φ are the usual polar and
azimuthal angles. Consider a rotation about the z-axis by the angle •; then

’ ←
’ ←←
S = R (•) S R ’1 (•)

’ ←
’ ’1 ←
’ ←’
= ∆ ⊥ k R (•) [← r (ω)] R ’1 (•) ∆ ⊥ k

’ ’1 ←

= ∆ ⊥ k [← r (ω)] ∆ ⊥ k ,
’ (B.148)

where k is the rotated unit vector and we have used the invariance of ← r under

’ ←’
rotations around the z-axis. The matrices S and S are related by a similarity
transformation, so they have the same eigenvalues for any •. The choice • = ’φ
Classical electrodynamics

e¬ectively sets φ = 0, so the eigenvalues of S can only depend on θ, the angle
between k and the distinguished axis. Setting φ = 0 simpli¬es the calculation and the
two indices of refraction are given by

n2 = ⊥, (B.149)

n2 = . (B.150)
⊥ (1 ’ cos 2θ) + z (1 + cos 2θ)

The phase velocity c/no , which is independent of the direction of k, characterizes
the ordinary wave, while the phase velocity c/ne , which depends on θ, describes the
propagation of the extraordinary wave. The corresponding refractive indices no and
ne are respectively called the ordinary and extraordinary index.

B.5.4 Nonlinear optics
Classical nonlinear optics (Boyd, 1992; Newell and Moloney, 1992) is concerned with
the propagation of classical light in weakly nonlinear media. Most experiments in
quantum optics involve substances with very weak magnetic susceptibility, so we will
simplify the permeability tensor to µij (ω) = µ0 δij . On the other hand, the coupling to
the electric ¬eld can be strong, if the ¬eld is nearly resonant with a dipole transition
in the constituent atoms. In such cases, the relation between the polarization and
the ¬eld is not linear. In the simplest situation, the response of the atomic dipole
to the external ¬eld can be calculated by time-dependent perturbation theory, which
produces an expression of the form (Boyd, 1992, Chap. 3)

P (r, t) = P (1) (r, t) + P NL (r, t) , (B.151)

where the nonlinear polarization

P NL (r, t) = P (2) (r, t) + P (3) (r, t) + · · · (B.152)

contains the higher-order terms in the perturbation expansion and de¬nes the nonlin-
ear constitutive relations. The transverse electric ¬eld describing radiation satis¬es
eqn (B.103), and”after using eqn (B.151) and imposing the convention that E always
means the transverse part, E ⊥ ”this can be written as

1 ‚2 ‚ 2 ⊥(1) ‚ 2 ⊥NL
∇ E ’ 2 2 E ’ µ0 2 P = µ0 2 P
. (B.153)
c ‚t ‚t ‚t
The interesting materials are often crystals, so scalar relations between the polar-
ization and the ¬eld must be replaced by tensor relations for anisotropic media. In a
microscopic description, the polarization P is the sum over the induced dipoles in each
atom, but we will use a coarse-grained macroscopic treatment that is justi¬ed by the
presence of many atoms in a cubic wavelength. Thus the macroscopic susceptibilities
are proportional to the density, nat , of atoms, i.e. χ(n) = nat γ (n) , where γ (n) is the
nth-order atomic polarizability. In addition to coarse graining, we will assume that the
polarization at r only depends on the ¬eld at r, i.e. the susceptibilities do not exhibit
Macroscopic Maxwell equations

the property of spatial dispersion discussed in Appendix B.5.1. For the crystals used in
quantum optics spatial dispersion is weak, so this assumption is justi¬ed in practice.
In the time domain the nth-order polarization is given by

(n) (n)
Pi dt1 · · · dtn χij1 j2 ···jn (t ’ t1 , t ’ t2 , . . . , t ’ tn )
(r, t) = 0

— Ej1 (r, t1 ) · · · Ejn (r, tn ) , (B.154)
where χij1 j2 ···jn („1 , „2 , . . . , „n ) is real and symmetric with respect to simultaneous
permutations of the time arguments „p and the corresponding tensor indices jp . The
corresponding frequency-domain relation is
n n
(n) (n)
Pi ν’
(r, ν) = 2πδ νp χij1 j2 ···jn (ν1 , . . . , νn )

q=1 p=1
— Ej1 (r, ν1 ) · · · Ejn (r, νn ) , (B.155)

n n
(n) (n)
χij1 j2 ···jn (ν1 , . . . , νn ) = d„q exp i νp „p χij1 j2 ···jn („1 , „2 , . . . , „n ) . (B.156)
q=1 p=1

This notation agrees with one of the conventions (Newell and Moloney, 1992, Chap. 2d)


. 25
( 27)